1
JEE Main 2022 (Online) 25th June Morning Shift
+4
-1

Let $$\overrightarrow a = {a_1}\widehat i + {a_2}\widehat j + {a_3}\widehat k$$ $${a_i} > 0$$, $$i = 1,2,3$$ be a vector which makes equal angles with the coordinate axes OX, OY and OZ. Also, let the projection of $$\overrightarrow a$$ on the vector $$3\widehat i + 4\widehat j$$ be 7. Let $$\overrightarrow b$$ be a vector obtained by rotating $$\overrightarrow a$$ with 90$$^\circ$$. If $$\overrightarrow a$$, $$\overrightarrow b$$ and x-axis are coplanar, then projection of a vector $$\overrightarrow b$$ on $$3\widehat i + 4\widehat j$$ is equal to:

A
$$\sqrt 7$$
B
$$\sqrt 2$$
C
2
D
7
2
JEE Main 2022 (Online) 24th June Evening Shift
+4
-1

Let $$\widehat a$$ and $$\widehat b$$ be two unit vectors such that $$|(\widehat a + \widehat b) + 2(\widehat a \times \widehat b)| = 2$$. If $$\theta$$ $$\in$$ (0, $$\pi$$) is the angle between $$\widehat a$$ and $$\widehat b$$, then among the statements :

(S1) : $$2|\widehat a \times \widehat b| = |\widehat a - \widehat b|$$

(S2) : The projection of $$\widehat a$$ on ($$\widehat a$$ + $$\widehat b$$) is $${1 \over 2}$$

A
Only (S1) is true.
B
Only (S2) is true.
C
Both (S1) and (S2) are true.
D
Both (S1) and (S2) are false.
3
JEE Main 2022 (Online) 24th June Morning Shift
+4
-1

Let $$\widehat a$$, $$\widehat b$$ be unit vectors. If $$\overrightarrow c$$ be a vector such that the angle between $$\widehat a$$ and $$\overrightarrow c$$ is $${\pi \over {12}}$$, and $$\widehat b = \overrightarrow c + 2\left( {\overrightarrow c \times \widehat a} \right)$$, then $${\left| {6\overrightarrow c } \right|^2}$$ is equal to :

A
$$6\left( {3 - \sqrt 3 } \right)$$
B
$$3 + \sqrt 3$$
C
$$6\left( {3 + \sqrt 3 } \right)$$
D
$$6\left( {\sqrt 3 + 1} \right)$$
4
JEE Main 2021 (Online) 31st August Evening Shift
+4
-1
Let $$\overrightarrow a ,\overrightarrow b ,\overrightarrow c$$ three vectors mutually perpendicular to each other and have same magnitude. If a vector $${ \overrightarrow r }$$ satisfies.

$$\overrightarrow a \times \{ (\overrightarrow r - \overrightarrow b ) \times \overrightarrow a \} + \overrightarrow b \times \{ (\overrightarrow r - \overrightarrow c ) \times \overrightarrow b \} + \overrightarrow c \times \{ (\overrightarrow r - \overrightarrow a ) \times \overrightarrow c \} = \overrightarrow 0$$, then $$\overrightarrow r$$ is equal to :
A
$${1 \over 3}(\overrightarrow a + \overrightarrow b + \overrightarrow c )$$
B
$${1 \over 3}(2\overrightarrow a + \overrightarrow b - \overrightarrow c )$$
C
$${1 \over 2}(\overrightarrow a + \overrightarrow b + \overrightarrow c )$$
D
$${1 \over 2}(\overrightarrow a + \overrightarrow b + 2\overrightarrow c )$$
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